An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data

In this paper, a finite volume element (FVE) method is considered for spatial approximations of time fractional diffusion equations involving a Riemann-Liouville fractional derivative of order α ∈ (0, 1) in time. Improving upon earlier results [Karaa et al., IMA J. Numer. Anal. 37 (2017) 945–964], error estimates in L2 (Ω)- and H1 (Ω)-norms for the semidiscrete problem with smooth and mildly smooth initial data, i.e., v ∈ H2(Ω) ∩ H01(Ω) and v ∈ H01(Ω) are established. For nonsmooth data, that is, v ∈ L2 (Ω), the optimal L2 (Ω)-error estimate is shown to hold only under an additional assumption on the triangulation, which is known to be satisfied for symmetric triangulations. Superconvergence result is also proved and as a consequence, a quasi-optimal error estimate is established in the L∞ (Ω)-norm. Further, two fully discrete schemes using convolution quadrature in time generated by the backward Euler and the second-order backward difference methods are analyzed, and error estimates are derived for both smooth and nonsmooth initial data. Based on a comparison of the standard Galerkin finite element solution with the FVE solution and exploiting tools for Laplace transforms with semigroup type properties of the FVE solution operator, our analysis is then extended in a unified manner to several time fractional order evolution problems. Finally, several numerical experiments are conducted to confirm our theoretical findings.

We study the space-time finite element discretizations for time fractional parabolic integro-differential equations in a bounded convex polygonal domain in $${\mathbb {R}}^d (d=1,2,3)$$ . Both spatially semidiscrete and fully discrete finite element approximations are considered and analyzed. We use piecewise linear and continuous finite elements to approximate the space variable whereas the time discretization uses two fully discrete schemes based on the convolution quadrature, namely the backward Euler and the second-order backward difference. For the spatially discrete scheme, optimal order a priori error estimates are derived for smooth initial data, i.e., when $$u_0\in H_0^1(\varOmega )\cap H^2(\varOmega )$$ . Moreover, for the homogeneous problem, almost optimal error estimates for positive time are established for nonsmooth initial data, i.e., when the initial function $$u_0$$ is only in $$ L^2(\varOmega )$$ . The error estimates for the fully discrete methods are shown to be optimal in time for both smooth and nonsmooth initial data under the specific choice of the kernel operator in the integral. Finally, we provide some numerical illustrations to verify our theoretical analysis.

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

Fully discrete approximation methods are studied for parabolic boundary value problems evolving from nonsmooth (and noncompatible) initial data. Galerkin methods are used for the spatial discretization, and single step or backwards differentiation multistep methods are used in time. The parabolic equation is assumed to be linear, but the coefficients are allowed to depend on space and time. An error estimate is given for the combined space-time discretization scheme which contains the expected optional order space and time error contributions. However, another term appears which limits the accuracy of the method to second order (or fractionally better) in time if the time stepping method is of third or higher order. This latter term arises from the time dependence of the equation’s highest order coefficients, and hence does not appear in analyses of problems with only spatially dependent coefficients for which optimal order estimates are known to hold. It is shown through more detailed analysis of a specific problem, and the formulation of a lower error bound, that the error estimate is indeed sharp. Thus, it follows that standard higher order time stepping schemes cannot be expected to perform well for parabolic problems with time dependent coefficients in situations when nonsmooth initial data are encountered.

Two higher order time stepping methods for solving subdiffusion problems are studied in this paper. The Caputo time fractional derivatives are approximated by using the weighted and shifted Grünwald–Letnikov formulae introduced in Tian et al. (Math Comput 84:2703–2727, 2015). After correcting a few starting steps, the proposed time stepping methods have the optimal convergence orders O(k^2) and O(k^3), respectively for any fixed time t for both smooth and nonsmooth data. The error estimates are proved by directly bounding the approximation errors of the kernel functions. Moreover, we also present briefly the applicabilities of our time stepping schemes to various other fractional evolution equations. Finally, some numerical examples are given to show that the numerical results are consistent with the proven theoretical results.

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

We derive rigorous error bounds for exponential Runge–Kutta discretizations of parabolic equations with nonsmooth initial data. Our analysis is carried out in the framework of abstract semilinear evolution equations, allowing for operators with non-dense domains. The results provide a foundation for establishing error estimates for nonsmooth data in prototypical problems such as the Allen-Cahn and Burgers equations. Furthermore, we apply these estimates to the analysis of split exponential integrators, yielding convergence results expressed explicitly in terms of the prescribed data.

ABSTRACTIn this paper, we will study solution existence and its stability for hybrid fractional DE with fractional integral, fractional differential derivative and Φp-operator in Caputo sense. Our problem deals with two basic types of fractional order derivatives, that is, Riemann-Liouville derivative of order and Caputo fractional derivative of order where and We will transform the problem into an integral equation by using Green function and examine whether it is increasing or decreasing and positive or negative function. Some fixed point theorems (Krasnoselskii Theorem) are utilized for the existence of a positive solution (EPS). Addition to studying HU-stability technique for our suggested problem. An example is included to apply the results.

Abstract. We study spatially semidiscrete and fully discrete finite volume element methods for the homogeneous heat equation with homogeneous Dirichlet boundary conditions and derive error estimates for smooth and nonsmooth initial data. We show that the results of our earlier work [Math. Comp. 81 (2012), 1–20] for the lumped mass method carry over to the present situation. In particular, in order for error estimates for initial data only in L2 to be of optimal second order for positive time, a special condition is required, which is satisfied for symmetric triangulations. Without any such condition, only first order convergence can be shown, which is illustrated by a counterexample. Improvements hold for triangulations that are almost symmetric and piecewise almost symmetric.

Uniform [formula omitted] behavior of the first-order interpolant quadrature scheme for some partial integro-differential equations

High-order time-stepping methods for two-dimensional Riesz fractional nonlinear reaction–diffusion equations

In this article, a meshfree method for the numerical solution of conversation law equations is considered. Some problems which have shock such as advection problems are not properly solved by radial basis function collocation meshfree method. Therefore, we use the integrated radial basis function (IRBF) method for some of these problems. In the current study, the governing models have been discretized by IRBF technique in the spatial direction and by finite difference approximation for time variable. This converts the main problem to a system of nonlinear ordinary differential equations (ODEs). Furthermore, the obtained ODEs will be solved by Runge–Kutta technique. This is the meshless method of lines technique. Numerical examples indicate the acceptable accuracy, proficiency and easy implementation of the presented method.

The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order α ∈ (0, 1). After using quasi-boundary value method to regularize the ‘mildly’ ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. The theoretical results are useful to balance discretization parameters, regularization parameter and noise level. Numerical examples are presented to illustrate the theoretical results.

This paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one, reflected at its infimum. First we derive the forward equation using the theory of sun-dual semigroups. The resulting forward equation is a boundary value problem on the positive half-line that involves a negative Riemann-Liouville fractional derivative in space, and a fractional reflecting boundary condition at the origin. Then we apply numerical methods to explicitly compute the transition density of this space-inhomogeneous Markov process, for any starting point, to any desired degree of accuracy. Finally, we discuss an application to fractional Cauchy problems, which involve a positive Caputo fractional derivative in time.

This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results.